Encoding method for the compression of a video sequence

ABSTRACT

The invention relates to an encoding method for the compression of a video sequence by means of a tridimensional wavelet transform. This method is based on a hierarchical subband encoding process leading to transform coefficients constituting a hierarchical pyramid. A spatio-temporal orientation tree, in which the roots are formed with the pixels of the approximation subband and the offspring of each of these pixels is formed with the pixels of the higher subbands, defines the spatio-temporal relationship inside said pyramid. According to the invention, the initial subband structure of the wavelet transform is preserved, in the encoding process, by scanning the subbands one after the other in an order that respects the parent-offspring dependencies formed in the tree. Moreover, flags “off/on” are added to each coefficient of the tree in view of a progressive transmission of the most significant bits of the coefficients, at least one of them describing the state of a set of pixels and at least another one describing the state of a single pixel.

FIELD OF THE INVENTION

The present invention relates to an encoding method for the compressionof a video sequence divided in groups of frames decomposed by means of athree-dimensional (3D) wavelet transform leading to a given number ofsuccessive resolution levels corresponding to the decomposition levelsof said transform, said method being based on a hierarchical subbandencoding process leading from the original set of picture elements(pixels) of each group of frames to transform coefficients constitutinga hierarchical pyramid, and a spatio-temporal orientation tree—in whichthe roots are formed with the pixels of the approximation subbandresulting from the 3D wavelet transform and the offspring of each ofthese pixels is formed with the pixels of the higher subbandscorresponding to the image volume defined by these root pixels—definingthe spatio-temporal relationship inside said hierarchical pyramid.

BACKGROUND OF THE INVENTION

The expansion of multimedia applications is now making the scalabilityone of the most important functionalities of video compression schemes.Scalability allows delivering multiple levels of quality or spatialresolutions/frame rates in an embedded bitstream towards receivers withdifferent requirements and encoding capabilities. Current standards likeMPEG-4 have implemented scalability in a predictive DCT-based frameworkthrough additional high-cost layers. More efficient solutions based on athree-dimensional wavelet decomposition followed by a hierarchicalencoding of the spatio-temporal trees like the Set Partitioning InHierarchical Trees algorithm (SPIHT) have been recently proposed as anextension of still image coding techniques (the original SPIHT algorithmis described for instance in “A new, fast, and efficient image codecbased on set partitioning in hierarchical trees”, by A. Said and W. A.Pearlman, IEEE Transactions on Circuits and Systems for VideoTechnology, vol.6, no 3, June 1996, pp.243–250, and the extension ofthis algorithm to the 3D case is described for instance in “An embeddedwavelet video coder using three-dimensional set partitioning inhierarchical trees (SPIHT)”, B. J. Kim and W. A. Pearlman, Proceedingsof Data Compression Conference, Mar. 25–27, 1997, Snowbird, Utah, USA,pp.251–260). The 3D wavelet decomposition provides a natural spatialresolution and frame rate scalability, while the in-depth scanning ofthe obtained coefficients in the hierarchical trees and the bitplaneencoding lead to the desired quality scalability with a high compressionratio.

The SPIHT algorithm is based on a key concept: a partial sorting of thecoefficients according to a decreasing magnitude, and the prediction ofthe absence of significant information across scales of the waveletdecomposition by exploiting self-similarity inherent in natural images.This means that if a coefficient is insignificant at the lowest scale ofthe wavelet decomposition, the coefficients corresponding to the samearea at the other scales have a high probability to be insignificanttoo. Basically, the SPIHT is an iterative algorithm that consists incomparing a set of pixels corresponding to the same image area atdifferent resolutions with a value called “level of significance” fromthe maximal significance level found in the spatio-temporaldecomposition tree down to 0. For a given level, or bitplane, two passesare carried out: the sorting pass, which looks for zero-trees orsub-trees and sorts insignificant and significant coefficients, and therefinement pass, which sends the precision bits of the significantcoefficients. The SPIHT algorithm examines the wavelet coefficients fromthe highest level of the decomposition to the lowest one. Thiscorresponds to first considering the coefficients corresponding toimportant details located in the smallest scale subbands, withincreasing resolution, then examining the smallest coefficients, whichcorrespond to fine details. This justifies the “hierarchical”designation of the algorithm: the bits are sent by decreasing importanceof the details they represent, and a progressive bitstream is thusformed.

A tree structure, called spatial (or spatio-temporal in the 3D case)orientation free, defines the spatial (or spatio-temporal) relationshipinside the hierarchical pyramid of wavelet coefficients. The roots ofthe trees are formed with the pixels of the approximation subband at thelowest resolution (“root” subband), while the pixels of the highersubbands corresponding to the image area (to the image volume, in the 3Dcase) defined by the root pixel form the offspring of this pixel. In the3D version of the SPIHT algorithm, each pixel of any subband but theleaves has 8 offspring pixels, and each pixel has only one parent. Thereis one exception at this rule: in the root case, one pixel out of 8 hasno offspring. The following notations describe the parent-offspringrelationship, an illustration of these dependencies being given in Fig.l(three-dimensional case) where the notations are the following:TF=temporal frame, TAS=temporal approximation subband, CFTS coefficientsin the spatio-temporal approximation subbands (or root coefficients),TDS.LRL=temporal detail subband at the last resolution level of thedecomposition, and TDS.HR=temporal detail subband at higher resolution:

-   -   O(x,y,z): set of coordinates of the direct offspring of the node        (x,y,z);    -   D(x,y,z): set of coordinates of all descendants of the node        (x,y,z);    -   H(x,y,z): set of coordinates of all spatio-temporal orientation        tree roots (nodes in the highest pyramid level: spatio-temporal        approximation subband);    -   L(x,y,z)=D(x,y,z)−O(x,y,z).

The SPIHT algorithm makes use of three lists: the LIS (list ofinsignificant sets), the LIP (list of insignificant pixels), and the LSP(list of significant pixels). In all these lists, each entry isidentified by a coordinate (x,y,z). In the LIP and LIS, (x,y,z)represents a unique coefficient, while in the LIS it represents a set ofcoefficients D(x,y,z) or L(x,y,z), which are sub-trees of thespatio-temporal tree. To differentiate between them, the LIS entry is oftype A if it represents D(x,y,z), and of type B if it representsL(x,y,z). During the first pass (sorting pass), all the pixels of theLIP are tested and those that become significant are moved to the listLSP. Similarly, the sets of the LIS that become significant are removedfrom the list LIS and split into subsets that are placed at the end ofthe LIS and will be each examined in turn. The LSP contains the list ofsignificant pixels to be “refined”: the n^(th) bit of the coefficient issent if this one is significant with respect to the level n.

The SPIHT approach is designed to provide quality scalability associatedwith a high compression ratio. However, scalability in temporal orspatial resolutions cannot be obtained with this coding strategy withoutmodifications. To improve the global compression rate of the videocoding system, it is usually advised to add an arithmetic encoder to thezero-tree encoding module. In other approaches, the arithmetic codinguses pertinent contexts directly applied to the subbands for losslessimage compression. Most of the time, the hierarchical and arithmeticcoding modules are considered separately. To efficiently combine them ina single coding system, some modifications have to be performed on theoriginal SPIHT algorithm.

To make the arithmetic coding efficient, it is very important to captureall the information that may have some influence on the current pixeland particularly the information related to neighbouring pixels. Thisinformation is represented by its context. The in-depth search performedwhen scanning for zero-trees does not exploit the redundancy insidesubbands and makes harder the determination of a relevant context forthe arithmetic coding. The manipulation of the lists LIS, LIP, LSPconducted by a set of logical conditions makes the order of pixelscanning hardly predictable. The pixels belonging to the same 3Doffspring tree but coming from different spatio-temporal subbands areencoded and put one after the other in the lists, which has for effectto mix the pixels of foreign subbands. Thus, the geographicinterdependencies between pixels of the same subband are lost. Moreover,since the spatio-temporal subbands result from temporal or spatialfiltering, the frames are filtered along privileged axes that give theorientation of the details. This orientation dependency is also lostwhen the SPIHT algorithm is applied, because the scanning does notrespect the geographic order.

Furthermore, the bits resulting from the examination of the lists LIS,LIP, LSP and the signs of the coefficients have quite differentstatistical properties. The relevant contexts for one list can betotally different from another. For example, as the LIP represents theset of insignificant pixels, it is reasonable to suppose that if a pixelis surrounded by insignificant pixels, it has great chance to beinsignificant too, but this supposition seems bolder for the LSP: itcannot be necessarily deduced that the refinement bit of an examinedpixel is one (resp. zero) if the refinement bits of its neighbours areones (resp. zeros) at a certain level of significance.

Faced with the difficulties to add an entropy coding stage to the SPIHTalgorithm, the documents that relate such an implementation are quiteelusive, or even skeptical about the efficiency of the proposedsolutions. Most of the time, the hierarchical coding methods and thecontext-based lossless image compression methods are confronted in thecase of still pictures. In the case of a video sequence, the SPIHTencoding strategy is very efficient to provide a fully qualityprogressive bitstream with a high compression rate, but the hierarchicalstructure used in said strategy however does neither facilitate theinsertion of a context-based adaptive arithmetic coding nor thefunctionality of spatial or temporal resolution scalability, which isstrongly required by emerging multimedia applications.

SUMMARY OF THE INVENTION

It is therefore an object of the invention to propose a new strategy forencoding the spatio-temporal wavelet coefficients, inspired from the3D-SPIHT, but which allows a better context selection while allowing toobtain a spatial or temporal resolution scalability in the codingscheme.

To this end, the invention relates to an encoding method such as definedin the introductive part of the description and which is moreovercharacterized in that:

-   -   (A) the initial subband structure of the 3D wavelet transform is        preserved by scanning the subbands one after the other in an        order that respects the parent-offspring dependencies formed in        said spatio-temporal tree;    -   (B) flags “off/on” are added to each coefficient of the        spatio-temporal tree in view of a progressive transmission of        the most significant bits of the coefficients, these flags being        such that at least one of them describes the state of a set of        pixels and at least another one describes the state of a single        pixel.

Although the use of lists LIS, LIP and LSP in the original SPIHTalgorithm facilitates the classification task, it is an obstacle to ageographic organization of the coefficients. By using the presenttechnique, the initial subband structure of the 3D wavelet transform ispreserved, and a flag added to each coefficient indicates to which listLIS, LIP or LSP this coefficient belongs. Thus, the scanning of thelists is replaced by a subband scanning and a flag interpretation: thehierarchical and logical organization of the SPIHT is preserved, and inthe same time moving a coefficient from a list to another is “virtually”done by changing its flag. The interest of this “virtual moving” is thatthe order of reading is not dependent of the changes performed by thelogic of the SPIHT algorithm, which is particularly interesting for therefinement pass, since the refinement bits constitute the greatest partof the bitstream.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will now be described, with reference to theaccompanying drawings in which:

FIG. 1 gives examples of parent-offspring dependencies in the 3D case,in the spatio-temporal orientation tree;

FIG. 2 illustrates the hierarchy of the subbands in said spatio-temporaltree;

FIG. 3 shows a spatially-driven scanning of the spatio-temporal tree;

FIG. 4 depicts a bitstream organization made possible by the ordered 3DSPIHT;

FIG. 5 shows a temporally-driven scanning of the spatio-temporal tree,and

FIG. 6 depicts the structure of the bitstream obtained with saidscanning;

FIG. 7 illustrates a combination of SNR, spatial and temporalscalabilities using the spatially-driven scanning strategy;

FIG. 8 shows the hierarchical organization of the bitstream withoutresolution flags.

DETAILED DESCRIPTION OF THE INVENTION

In the considered method, the whole spatio-temporal tree is fullyscanned for each new bitplane. At the end of the first bitplane, all theoffspring dependencies of the 3D volume have been evaluated. This firstscanning is therefore quite critical and must absolutely respect thecalculation order of the offspring dependencies described in FIG. 2,where the notations are the following: SA=spatial (s) axes, TA=temporal(t) axis, R=roots, FC=first children, SC=second children, and TC=thirdchildren. According to the invention, the proposed algorithm scans thesubbands one after the other in an order that respects theparent-offspring relationships. At least two different flags, andpreferably four, are added to the coefficients of the spatio-temporaltree:

-   A) at least one, and preferably two of them describe the state of a    set (trees or subtrees):    -   DIRECT_(—SET)        _(—INSIG (or FS1) if D(x,y,z) is still insignificant;)    -   UNDIRECT_(—SET)        _(—INSIG (or FS2) if L (x,y,z) is still insignificant;)-   B) at least another one, and preferably the two other ones describe    the state of a single pixel:    -   SIG (or FP3) if the current pixel is significant;    -   INSIG (or FP4) if it is not significant, or if its significance        is to be analyzed (put by default to the pixels that are not        included in a zero-tree).        The main steps of the algorithm implemented in the present        method are:-   1. Initialization:    -   Put flag FP4 to all the coefficients of the lowest        spatio-temporal subband;    -   Put flag FS1 to 7 over 8 coefficients of the lowest        spatio-temporal subband.-   2. Calculate and output MSL (the maximum significance level found in    the spatio-temporal decomposition tree).-   3. From n=MSL down to 0, do a full exploration of the    spatio-temporal tree (two main approaches are possible, as described    in the following paragraph: spatially-driven resolution scalability,    and temporally-driven resolution scalability), with, for each    coefficient (x,y,z) of the spatio-temporal tree, the following    actions:

a) set significance: 1) if flag FS1 is “on”, then output = S_(n)(D(x,y,z)). if S_(n) (D(x,y,z)) = 1, then: for each (x′,y′,z′) εO(x,y,z), put flag FP4; remove flag FS1 from (x,y,z); if L(i,j) ≠ Ø,then put flag FS2. 2) if flag FS2 is “on”, then output = S_(n)(L(x,y,z)).  if S_(n) (L(x,y,z)) = 1, then: for each (x′,y′,z′) εO(x,y,z), put flag FS1; remove flag FS2 from (x,y,z). b) pixelsignificance: 1) if flag FP3 is on, then output = the n^(th) bit of(x,y,z). 2) if flag FP4 is on, then output = S_(n) (x,y,z).  ifS_(n)(x,y,z) = 1, then: put flag FP3 on; output sign (x,y,z); removeflag FP4.

The frames are filtered along privileged axes (spatial or temporal) thatgive the orientations of the details. These orientations can be bettertaken into account by scanning the subband along the same directions.Using this algorithm, there are then two main ways of exploring thespatio-temporal volume of coefficients depending on the chosenprivileged orientation chosen, which may be either the spatial or thetemporal axis. Consequently, two types of “multi-scalable” bitstreamsmay be obtained, one leaded by the spatial resolution, the second by thetemporal resolution:

-   -   (A) spatially-driven resolution scalability:

For each bitplane, the tree scanning is spatially oriented, since inthis scheme the spatial resolutions are fully explored one after theother as shown in FIG. 3. Inside each spatial scale, all the temporalresolutions are successively scanned. In other words, the temporalfrequency is higher than the spatial one. In order to have thepossibility to skip some part of the bitstream, it is necessary tointroduce resolution flags in the bitstream. The scanning strategy leadsto a video bitstream organized as indicated in FIG. 4, where the lines sand t correspond respectively to spatial and temporal decompositionlevels (SDL and TDL), the black flags are flags separating twobitplanes, and the grey flags are flags separating two spatialdecomposition levels.

-   -   (B) temporally-driven resolution scalability:

For each bitplane, the tree scanning is temporally oriented, since inthis scheme the temporal resolutions are fully explored one after theother, as shown in FIG. 5. Inside each temporal scale, all the spatialresolutions are successively scanned and therefore all the spatialfrequencies are available. This scanning strategy leads to a videobitstream organized as indicated in FIG. 6, to be compared with FIG. 4(the grey flags are now flags separating two temporal decompositionlevels). In both cases, the three types of scalability (temporal,spatial resolution, SNR) are obtained:

-   -   SNR scalability is still available since the spatio-temporal        scanning is inserted in a bitplane iterative loop;    -   temporal and spatial scalability are provided respectively with        t_(max) possible frame rates and s_(max) possible display sizes        (t=1 to 4 and s=1 to 4 in the described examples), with t=1        corresponding to the minimum frame rate_(min), and s=1        corresponding to the minimum display size.        An example of selective decoding is illustrated in FIG. 7.

The advantages of the implementation of the method according to theinvention are the following:

(A) improvement of contexts: thanks to the fixed subband scanning andthe recognition of the flags, it is possible to reestablish a coherentgeographic context for each model (indeed, the SPIHT algorithm aims atreducing the redundancy between subbands of different scales, but itdoes not really take into account the geographic redundancy, unlike thecontext-based coding approaches), which is particularly interesting forthe coding of the significant pixels and their refinement bits (for thesignificant pixels, thanks to the algorithm proposed, the sameefficiency as with the SPIHT algorithm can be reached, and the rules ofconstruction of the context are quite simple). This method betterexploits the neighbouring influence on the current pixel than thosewhich combine classical SPIHT algorithm and entropy coding, and leads toa “natural” context, directly issued from the transformed image, inconformity with the bitplane approach, and not from the bits resultingfrom the original SPIHT algorithm in the refinement passes. Said methodshould improve the compression rate, as the context is really related tothe bit being encoded, but, as it scans all the subbands entirely, thecomputation time for the first levels is greater than with the formermethod.

(B) trade-off between multi-scalability and bitstream overload : thepossibility to reconstruct video sequences with the desired frame rateand display sizes by extracting the corresponding fragments of thebitstream is an attractive concept, but it is obtained at the expense ofcoding efficiency for two main reasons:

-   (a) the bitstream fragments related to a particular spatial or    temporal resolution need to be separated by a flag to make jumps    possible. With the two scalability schemes described above, on the    examples given, at least four separators are needed per bitplane,    and up to 12 bitplanes are currently used to encode the wavelet    coefficients.-   (b) the context calculation of the adaptive arithmetic coding module    must be reinitialized at the beginning of each new bitplane to    ensure that any bitstream fragment will be processed at the decoder    side in exactly the same conditions as at the encoder side.    Therefore the multiplication of separators will unavoidably reduce    the length of the consecutive bit sequences encoded by the    arithmetic coding module and makes harder the probability    estimation. However, as the subbands can be considered as non or    partially stationary sources, this apparent drawback could be a    quality.

A trade-off must be found between full resolution scalability andarithmetic coding efficiency. To this end, an intermediate solution,which provides four levels of spatial and temporal scalabilities, isproposed. The minimal frame rate rate_(min) is always associated withthe minimal display size (S^(x)min, S^(y)min), to constitute the firstresolution level. As well 2*rate_(min) is combined with the display size(2*S^(x)min, 2*S^(y)min) etc. FIG. 8 illustrates this when there arefour resolution levels in the decomposition of the group of frames(GOF). All the combinations that were previously possible (16possibilities with 4 spatial levels and four temporal levels) are nowrestricted to four.

1. An encoding method for the compression of a video sequence divided ingroups of frames decomposed by means of a three-dimensional (3D) wavelettransform leading to a given number of successive resolution levelscorresponding to the decomposition levels of said transform, said methodbeing based on a hierarchical subband encoding process leading from theoriginal set of picture elements (pixels) of each group of frames totransform coefficients constituting a hierarchical pyramid, and aspatio-temporal orientation tree —in which the roots are formed with thepixels of the approximation subband resulting from the 3D wavelettransform and the offspring of each of these pixels is formed with thepixels of the higher subbands corresponding to the image volume definedby these root pixels—defining the spatio-temporal relationship insidesaid hierarchical pyramid, said encoding method comprising the steps of:(A) preserving the initial subband structure of the 3D wavelet transformby scanning the subbands one after the other in an order that respectsthe parent-offspring dependencies formed in said spatio-temporal tree;(B) adding “off/on” flags to each coefficient of the spatio-temporaltree in view of a progressive transmission of the most significant bitsof the coefficients, these flags being such that at least one of themdescribes the state of significance of a set of pixels and at leastanother one describes the state of significance of a single pixel,wherein two flags describe the state of significance of a set of pixelsand are, for each coefficient (x,y,z) of said spatio-temporal tree: FS1if D(x,y,z) is still insignificant; FS2 if L(x,y,z) is stillinsignificant; where D(x,y,z) is the set of coordinates of all thedescendants of the node (x,y,z) and L(x,y,z)=D(x,y,z)−O(x,y,z), withO(x,y,z) is the set of coordinates of the direct offspring of the node(x,y,z), and two flags describe the state of significance of a singlepixel and are: FP3 if the current pixel is significant; FP4 if it is notsignificant or if its significance is to be analyzed.
 2. The encodingmethod according to claim 1, wherein for each bitplane, the treescanning is spatially oriented, all the temporal resolutions beingsuccessively scanned inside each spatial scale and resolution flagsbeing introduced between any two spatial scales.
 3. The encoding methodaccording to claim 1, wherein for each bitplane, the tree scanning istemporally oriented, all the spatial resolutions being successivelyscanned inside each temporal scale and resolution flags being introducedbetween any two temporal scales.
 4. The encoding method according toclaim 1, wherein for each bitplane, an intermediate tree scanning isperformed, all the temporal and spatial resolutions of the same scalebeing jointly scanned and resolution flags being introduced between anytwo spatial/temporal scales.
 5. The encoding method according to claim1, wherein the exploration of the spatio-temporal tree, implemented insaid scanning order, includes, after an initialization step where theflag FP4 is put to all the coefficients of the lowest spatio-temporalsubband and the flag FS1 to 7 over 8 coefficients of said lowestspatio-temporal subband, and the maximum significance level MSL iscalculated, the following steps, carried out from the bitplane n =MSLdown to the bitplane n =0 and from the lowest subband resolution down tothe highest one: (a) a first set of tests related to the setsignificance; (1) if the flag FS1 is “on”, then output S_(n) (D(x,y,z)):if S_(n) (D(x,y,z)) = 1, then: for each (x′,y′,z′) in O(x,y,z), put flagFP4; remove flag FS1 from (x,y,z); if L(x,y,z) not empty, then put flagFS2. (2) if flag FS2 is “on”, then output S_(n) (L(x,y,z)): if S_(n)(L(x,y,z)) = 1, then: for each (x′,y′,z′) in O(x,y,z), put flag FS1;remove flag FS2 from (x,y,z). (b) a second set of tests related to thepixel significance: (1) if the flag FP3 is “on”, then output = the n-thbit of (x,y,z); (2) if the flag FP4 is “on”, then output S_(n) (x,y,z):if S_(n)(x,y,z) = 1, then: put flag FP3 “on”; output sign (x,y,z); andremove flag FP4.


6. The encoding method according to claim 1 further comprising the stepof: partially decoding the bitstream between two resolution flags,leading to a lower resolution/frame rate reconstructed video sequence.7. The encoding method according to claim 6, wherein the context usedfor the encoding of each bit related to the set significance in anarithmetic coding module is built using the bits of the same bitplane ofthe last scanned neighboring wavelet coefficients in the samespatio-temporal subband, these bits being the bits output during thefirst set of tests related to the set significance.
 8. The encodingmethod according to claim 6, wherein the context used for the encodingof each bit related to the pixel significance in an arithmetic codingmodule is built using the bits of the same bitplane of the last scannedneighboring wavelet coefficients in the same spatio-temporal subband,these bits being 1 if the neighboring coefficients are marked by an FP3flag and 0 if not.
 9. An encoding system for the compression of a videosequence divided in groups of frames decomposed by means of athree-dimensional (3D) wavelet transform leading to a given number ofsuccessive resolution levels corresponding to the decomposition levelsof said transform, said method being based on a hierarchical subbandencoding process leading from the original set of picture elements(pixels) of each group of frames to transform coefficients constitutinga hierarchical pyramid, and a spatio-temporal orientation tree —in whichthe roots are formed with the pixels of the approximation subbandresulting from the 3D wavelet transform and the offspring of each ofthese pixels is formed with the pixels of the higher subbandscorresponding to the image volume defined by these root pixels definingthe spatio-temporal relationship inside said hierarchical pyramid, saidencoding system comprising: a processor in communication with a memory,said processor executing code for: (A) preserving the initial subbandstructure of the 3D wavelet transform by scanning the subbands one afterthe other in an order that respects the parent-offspring dependenciesformed in said spatio-temporal tree; and (B) adding “off/on” flags toeach coefficient of the spatio-temporal tree in view of a progressivetransmission of the most significant bits of the coefficients, theseflags being such that at least one of them describes the state ofsignificance of a set of pixels and at least another one describes thestate of significance of a single pixel, wherein two flags describe thestate of significance of a set of pixels and are, for each coefficient(x,y,z) of said spatio-temporal tree: FS1 if D(x,y,z) is stillinsignificant; FS2 if L(x,y,z) is still insignificant; where D(x,y,z) isthe set of coordinates of all the descendants of the node (x,y,z) andL(x,y,z) =D(x,y,z) −O(x,y,z), with O(x,y,z) is the set of coordinates ofthe direct offspring of the node (x,y,z), and two flags describe thestate of significance of a single pixel and are: FP3 if the currentpixel is significant; FP4 if it is not significant or if itssignificance is to be analyzed.
 10. The system as recited in claim 9,wherein said code is stored in said memory.
 11. The system as recited inclaim 9, further comprising: an input/output device in communicationwith said processor and said memory.